Dispersion Lemmas Revisiteddownloads.hindawi.com/archive/1999/081341.pdfC6"2fl f [VuI2 o)(l+l/oe) dxdt. with ca > 0. Since (O/Oxl) (XJ(1 + L([n), the treatment ofthe term which

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Dispersion Lemmas Revisited

I. GASSERa’*, P. A. MARKOWICH b and B. PERTHAMEC

Universittt Hamburg, Institut f. Angew. Mathematik, Bundesstrafle 55, 20146 Hamburg, Germany,"b Johannes Kepler Universit{tt Linz, Institutf’r Analysis und Numerik, Altenberger StraJ3e 69, 4040 Linz, Austria,"

Ecole Normale Superieure, DMI, 45, rue d’Ulm, 75230 Paris Cedex 05, France

(Received 13 August 1997; In finalform 1 December 1998)

We investigate regularizing dispersive effects for various classical equations, e.g., theSchr6dinger and Dirac equations. After Wigner transform, these dispersive estimates arereduced to moment lemmas for kinetic equations. They yield new regularization results forthe Schr6dinger equation (valid up to the semiclassical limit) and the Dirac equation.

Keywords: Dispersion lemmas, Schr6dinger equation, Wigner equation, quantum-hydrodynamics,Dirac equation

1. INTRODUCTION

In this paper we investigate dispersive effects forvarious classical equations (Schr6dinger, Diracequations). We reduce these equations, after aWigner transform, to a kinetic type equation

Ot f + a(x, ) *x Vx f g. (1.1)

Then, we apply to (1.1) the same multiplier methodintroduced in [13] in the context ofpure Schr6dingerand kinetic equations. We show that it yieldsvarious new results. For the Schr6dinger equationwith potential, we obtain regularizing effects in

1/2LZt(Hx,loc) valid up to the semiclassical limit forpotentials V satisfying some kind of local one-sided

Lipschitz condition. These include Morawetz typeestimates and seem to be optimal in the semiclassi-cal limit. They are independent and complete thoseof Constantin and Saut [5] for merely boundedpotentials, or those of Ruiz and Vega [18]; Sj6grenand Sj61in [21]; Ben-Arzi and Klainerman [1].The case of the Dirac equation also yields new

results. They are obtained by reducing the Diracequation, written in the form ofa hyperbolic system,to a pseudo-differential equation iut P([D[)u witha real and radial symbol P(lcl). Then, we show thatthe dispersive effects for this equation can be provedthrough the classical Wigner transform and with a

modification of the usual multiplier (x/lxl). . Thisresult extends those of Colin [3] and also provides amore standard approach.

*Corresponding author.

365

366 I. GASSER et al.

We also would like to point out that our resultscombine and extend several ideas introduced inG6rard, Mauser, Markowich and Poupaud [9],concerning the approach to the Schr6dinger andDirac equations through Wigner transform, and inLions and Perthame [13] concerning the analogybetween dispersive and moment estimates. Thiskind of dispersive estimates where first establishedby Sj61in [20]; Constantin and Saut [4].

2. DISPERSION LEMMAS FORSCHRDINGER-TYPE PROBLEMS

The subsequent results will be based on the theoryof the Wigner-transform, for which we state hereonly the most basic (and important) properties.For an in-depth analysis we refer to [9, 12].LetfE S’(n; C). Then the Wigner transform of

f on the scale c > 0 is given by

w[f ](x,)

l (X q-- Cv)f(x-v)eivdv(27r)-- jR,J>

(2.1)

("-" denotes complex conjugation).Note that w[f]E S’(R x ; ). It is well

known that, in general, w[f] is not a positivemeasure.

For the following we use the definition

() (.x-+ f)({) .f (X)e-ix’dX

for the Fourier-transform on n. Its inverse isdenoted by

(x) --+x g)(x) (2rr)ng({)eixd{.

Trivially, we have

(f’_,vWe[f])(x,v)-f x+v f x-v

and, after a simple computation

We conclude

Jnwe[f ]d- lf (x) 2,

At first we consider the Schr6dinger equation onwith a given real valued potential V(x):

c2igut - Au + V(x)u, xn, t, (2.2a)

u(t O) ui on ". (2.2b)

It is verified by a simple calculation that w(x, , t) :=

wC[u(t)](x, ) satisfies the transport equation

w, + . VxW + 3[V]w o, (X,) 2n, ,(2.3a)

w(t-- O) w[ui] on 2n, (2.3b)

where O[V] is given by

(ID[V] f )(x,c)V(x+v) V(x-v)

(2)n JR C

(x-+-v)f(x--v) eivdv.

(2.4)

We shall now derive dispersive inequalities for theSchr6dinger equation (2.2) by employing a multi-plier technique on the Wigner equation (2.3).Although the subsequent results can also beobtained by using a multiplier technique directly

DISPERSION LEMMAS REVISITED 367

on the Schr6dinger equation, the ’detour’ via theWigner equation has two advantages. Firstly, ituses the connection between moment lemmas forkinetic equations and dispersion lemmas forSchr6dinger type equations employing a some-what intuitive multiplier on the kinetic level.Secondly it serves as a guideline for pseudodifferential equations with more general symbolswhere a direct multiplier approach without "ki-netic detour" would even be less intuitive than inthe Schr6dinger case.The first dispersion statement reads

THEOREM 2.1 For n >_ 2 let (-c2/2)A + V be ess-

entially self-adjoint and V(x)>_ a.e. in En, V E2 1/4Wl(En;R). Also assume uIED((--A+ V)

G L(’) and ([(x- x0)" VV(x)]+/lx- x0[)/( + v/V(x)) L(")foromxoe. Verare constants C independent ofxo and e such thatforall- oo < -l < 7"2 < O0 we have

(n- 2)

(2.5a)

0---5 vv () lul:

<_ Cn --A / V ui (n >_ 4),

n2 4n + 3c,

4(2.5c)

Here and in the sequel we denote by f+, f- thepositive and the negative part off, i.e., f= f+-f-.Proof We multiply the transport equation (2.3a)by (X-Xo/lX-Xo[). and integrate over

R x (q, -2). We obtain

(2.6)

where we used

lu(x, t)12 L w (x, , t)d

and set

s (x, t) f. w(x, ,The first term on the left hand side of (2.6) was

already computed in the reference [13] in terms ofthe wave function u. It is given by those terms onthe left hand side of (2.5), which do not involveV V(x). Using (2.4) we obtain

l[V] fd VxV (x)l f(x)] 2

which implies


Another simple calculation (using the definition ofthe Wigner-transform) gives

J (x, t) eIm (fi(x, t)Vu(x, t)) (2.7)

so that

x x0

x xo---" J (x, T)dx

(x-xo). Vu(x T)dxelm (x,T)[x_x0and

X X0

Ix xo" J (x, T)dx

<_ elVu(T)l_(1/2)x- xo u(T)Ix x01 (1/2)

Here we denote the (semi) norms on H(Nn) by

for a EN. Then we deduce from the aboveinequality

x x0

Ix xol" J (x, T)dx

e u(T)ll/2x xo u(T)x Xo[ 1/2

Now we consider the operators

XlR1 f -+ v-vf,x

1--1,...,n.

We compute

Cl2

only depends on the dimension n. Here we used[21() a,([l-l-n)" where "*" denotes the exten-

sion to 0. Since

Colg --+ i+r I>e ICo n+l

g(sc Co)dCo

is the Riesz-transformation (with index l) up to a

multiplicative constant, it is bounded on L2([n) (cf.[S, II, 4.2, Theorem 3]). Thus, it is also bounded on

Lz([Rn, Ild), where -n < a < n (cf. [S, II, 6.3])and

IRlf]l/2 <-- Clf 11/2

follows. We obtain

x x0

X xo------l" J (x, T)dx < CeIu(T)I2 (2.8)1/2

with a constant C, which only depends on n. Itremains to show the transport of v/-dul(t)[(1/2) bythe Schr6dinger flow with an e-independentconstant. We denote by S(t) the solution operatorof (2.2), i.e., S(t)ui is the solution u(t) of (2.2) attime t. We have

and we conclude

e2 1/4A / V)

e2 V,, 1/4VtEN.

Since V> 0 we have

e2 A) 1/4 e2 1/4 IL2(n)

for all D((- (e2/2)/k + V)(1/4)) (see [17], p. 44,prop. 9). Setting o u(t) gives

21/4 e2 )1/4

---A --]- V b/lL2(n)

Vtl


and by proceeding analogously

[Iw/4u(t)ll21l E2 )1/4

-5-zx+vV t [.

From (2.8) we derive:

X X0

Ix xo" J (x, T)dx

C ---/ nt- V un2(n)

and the assertion of Theorem (2.1) follows.

The next Theorem is concerned with the x-localized dispersion result for the Schr6dingerequation (2.2) where the singularity at x x0 in(2.5) is removed.

THEOREM 2.2 For n > let the assumptions on Vand ui of Theorem (2.1) hold. Then for 0 < c <_three is D D(n, c) > 0 independent of c such thatfor all-

The first term on the left hand side can be restatedin the wave-function u after a long and unpleasantcalculation. We obtain

where F(x)> 0 for 0 < c < (cf, [3]). Thus it hasthe lower bound

fl f [VuI2C6"2o) (l+l/oe)

dxdt.

with ca > 0. Since (O/Oxl) (XJ(1 +L([n), the treatment of the term which involvesJ in (2.10) is much simpler than of the one inTheorem 2.1.

f,2 l7ul 2

2(1 + x x0] ") l+l/.

dxdt

(1 / Ix x0 I) 1/dxdt

<_ D --g-A + V ui (2.9)

Proof As in [13] we multiply (2.3a) by (x.{/(1 +lxl)<*/% (we set Xo 0 to simplify theformulas) and obtain

7-2 x./l f2n" 7X( (1- i)i’ 1/O )wdxddt7-2 X" VV (x)

=[ x(J (x, j (x, )&.

(1 + Ixl) 1/T2

J(2.10)

Remark 2.1 Note that the term involving theelectric field VV in (3.5) can be estimated (due tothe assumptions of Theorem 2.1) by

-2 J x x0

IX XO VV (x)lul2dxdt

7-2n [(X-Xo)’VV(x)]-[x xo[u2dxdt

+ .(1 / v/V (x))lul2dxdt

and by interpolation:

7-2

fl n(I/V/V(X))[ul2dXdt< (T2--T1) Ilulll 2L2([n) / --’--A/ g Ul

2

L2(n)


Thus

)I-0- I-0, n--2

+ 4r]u(xo, t)] 2, n-3

x_xol3dx n 4

+ [(x- xo) V (x)]- ]

x A + V u

follows. The corresponding term in (2.9) is treatedanalogously.

Remark 2.2 The Theorems 2.1 and 2.2 containqualitatively different statements. Theorem 2.2

(which is more along the lines of the dispersionlemmas found in the literature [4, 5, 20, 22]) statesthat initial data with l(-e2/2 + V)(/4)UlJl(,)< yield Lo(t;Hl())-solutions u u(x, t)(under the stated assumptions on the potential).Theorem 2.1 (which holds for dimensions 2)instead has two main features. Firstly, the x-locality of Theorem 2.2 is replaced by a regulariza-tion property "away from the direction

u(x) x- Xo" and secondly, there is the singu-larity of the left-hand side of (2.5) at x x0 (c [3]).

Remark 2.3 A main point of both Theorems isthe precise (i.e., optimal) dependence of theestimates on the scaled Planck-constant e. Evenfor potentials VL() the approach of boot-strapping the dispersion results for the freeSchr6dinger equation by applying Duhamel’sformula (cf. [5]) gives results which are worse asfar as the dependence of e is concerned.

Remark 2.4 The class of admissible potentials inthe Theorems 2.1 and 2.2 is different from what wasconsidered in the literature [1, 5, 18, 21]. Note that,for example, the harmonic oscillator with potentialV(x) Ix] 2 is admissible, while important short-range potentials are excluded by the assumption

V-E L. This can somewhat be remedied at theexpense of the optimality of the appearing con-stants in dependence of c since the positivity of thepotential was only used to have a positive Hamilto-nian. For example, we can easily prove

COROLLARY 2.1 Assume that multiplication by[V(x)l is infinitesimally form bounded with respectto -zx ana tat ([x. x7V(x)]+/( + Ixl)l/)/(1 + (IV(x)l) (")for some o < 1. Alsolet uiE H(1/2)(n) and - < < 2 < . Thenthere is E E(n, ,, 2, ) > 0 such that

,1 + Ix)

fT2 [x" VV (x)]-- 2+( 1)/ lul2dxdt

(2.11)

Also, certain time dependent potentials can easilybe included in the theory.

Remark 2.5 (Nonlinear Schr6dinger equation)Consider the nonlinear IVP (n 2)"

c2

ieu,---- Au + h(lul2)u, xEBn, t[ (2.12a)

u(t O) bl in [Rn, (2.12b)

with the real enthalpy function h h(r), cr > 0.Using the multiplier (x-Xo)/(Ix-Xol)" for theWigner equation we obtain (after an integration byparts):

j;-EI )wdxddt+ (n- 1)

P(lUl2)dxdt

/R.Ix-x01(2.13)

where p’(cr)= h’(cr)r. Also, the conserved totalenergy is

E --- IX7ul2dx + H(lul2)dx’


with H’= h. Assume that H(cr)>- Kcr for someK > 0 and p(r)_> 0. Then, since the first term onthe left hand side of (2.13) is nonnegative and sincethe term on the right hand side is bounded by2E(t) + 211u(t)ll 2L2(.), we obtain

Zoo L p(lu(x, t))dxdtIX--XoI

<_ 2 fu lVuil2dx/2 on(lu,12)dx+ 2

JR[" ]Ull2dx"

(2.14)

This is a classical estimate (of Morawetz-type),which can be found in [6, 14]. Note that we alsocan bound the first term in (2.5) (involving thesingularity at x- x0) by a constant, which onlydepends on the dimension, the initial energy, and

Similar estimates [16] can also be obtained forthe Schr6dinger-Poisson problem (cf. [12, 15]) andfor Hartree-Fock equations [7].

Remark 2.6 (Quantum-hydrodynamics, cf. [8]) Forsome applications it is important to restate theestimates of Theorem 2.1 in terms of the macro-scopic densities

p(x, t) [u(x, t)12and J J(x, t) defined in (2.7). We obtain after alengthy calculation:

-2 [ [ (iji 2 ((x_xo)ilj)2)x plx- xol Ix- x0

+41x-xol Ix7l ((-)")

x x0-_0.vv(x)p ax(2.15)

+ 8ep(xo, t), n 3 dt

ix-xo

Q 2 )1/4 2

C -+V u

Remark 2.7 (Schr6dinger multiplier) Theorem 2.1can also be proven by multiplying the Schr6dingerequation by

x-xo n-

Ix- x0-----" 7 - 2 x- x01 ’taking real parts and integrating by parts (cf. [14]).

3. RADIAL FOURIER MULTIPLIERS

Finally we shall consider PDE’s with a real radialFourier multiplier P on

iu, P( DI)u, x E Nn, (3.1a)

u(t O) ui on ". (3.1b)

The case P(r)= r with u >_ 2 was analysed in [3]by using a specially constructed Wigner-typetransform. Here we shall employ the standardWigner transform (2.1) and allow a more generalclass of symbols (cf. [2] for local smoothingresults).

THEOREM 3.1that the map

Let P C2([0, oo); [) and assume

[1/2--+ 0"(, )d-l/2

sods

is one-to-one and onto from Nn to Nn for all Nn.Then the solution of (3.18) satisfies for all Xo Rn

{<__, n--2} 2 I12-, _> 3 (2)" I/"(11)----- 1’()12a’

(3.2a)


andfor O < a <_

(n_> 1).(3.2b)

Therefore, the work to express the left-hand side of(3.4) in terms of u has already been done [13] andgives rise to the left hand sides of (3.2). For thecomputation of the right-hand side we note that(3.3) gives

i,(, {, t) exp(-i( r(, )t)#,(, {)

Proof The evolution equation for the x-Fouriertransform (, , t) ’xW [u(t)] (we set ein (2.1)) reads:

(3.3)

(cf. [9]). We now define (r) by ’(r)= (r/(1 + r)(1/)) or, alternatively, g)(r)=r and com-pute

Next we test (3.3) against

We obtain

(3.4)

We now carry out the change of variables from {to cr cr(, {), denote D Idet(0cr/0{) and obtain

/ (. (-)2

i(({(-):. )2 (IxI-(C)D(C, or) -1

exp(-iff, aTt), #i(, (, or)) ),for l, 2. We compute

lim iC(Ixl)(ff)exp(-i(, crTt)

=-Ux- (r_+olim V(lx-aTI))=-2Cx (r+oolim sgn(rl)

I1e() -,

Also we obtain

With p(r)= r we have

and with ’(r)= r/(1 + r)(1/)

(((. )2(])((), 1)(,(

x.-/2o(’VX((l_i[v)l/)wd(dx"

where

(’ )-/: I + < ds,

fl/2 p,([ +3(’ )-l/


Since/3(if, )= O(I 1), we conclude

(0, or) .or

Io-I .(o,(o, I(o,

I1 P,( (o, or)I)"

Taking the limit 7-1 --- oo, r2-+ / oo in the right-hand side of (3.4) now gives

since cr (P’(IgI)/IgI)g for g 0.

It is an easy exercise to show that the conditionof Theorem 3.1 are satisfied for P(r)= r withu _> 2 (cf. [3], where also various other dispersionresults for 1-dim-problems can be found). Moregenerally we obtain

COROILAR 3.1 Let P E C2 ([0, oc); N), ((U(r)/r)’r C([0, oc); N) and P’(r) >_ r2l(p’(r)/r)’l forr >_ O. Then (3.19) (a), (b) hold for the solution of(3.18) (a), (b) when the "=" sign in (3.2a) is

replaced by "_<".

Clearly the same result holds if P’(r)<_-r2

I(P’(r)/r)’[ for all r > 0.Fourier-multipliers of the form P(r)= r for<_ u < 2 can be included in the theory be defining

approximate symbols P6(r) as follows"

d r

dr r + (5

A simple calculation shows that P6 satisfies theassumptions of Corollary 3.1. The limit process(5-+0 then proves that the assertions of theCorollary also hold for P(= lim_0P) r.A particularly interesting case is provided by

u- 1. We have for all x0 Rn(n > 2):

and consequently

X X0I< 21UI[ 20

I(x x0). ul ]x xol3dxdt

(3.5b)

Proof We set Q(r) (P’(r)/r) and compute where u is the solution of

00 [1/2 Q(I + <l)a la

/ f/2 Q’(lg / sl)

Therefore (0or/0{) _> 0 if Q(r) > rlQ’(r)l for r > 0. Ifwe replace P(r) by Pe(r) P(r)/ (1/2)(5r2 for (5 > 0we have (0ere/0{)>_ (5Id everywhere. By the globalimplicit function Theorem the map { --+ cr6(, {) is a

/ nhomeomorphism from R to R. Then the asser-tion follows by performing the limit (5-- 0 +.

iut- Du, x Nn, (3.6a)

u(t O) Ul. (3.6b)

Remark 3.1 (Dirac equation) We consider thefree Dirac equation in R3 for the spinorfieldU U(x,I)C4:

iUt R(D)U, x E 3, R (3.7a)

U(t- O)- UI on 3 (3.7b)


where

+ 70, :k=l 3

7t, 0,..., 4, are the 4 4 Dirac matrices. Theirelements are 0, 1, and they satisfy (cf. [11]):

70,(.,/0@). ,0.yk for k 1,2,3,

(7o)2-/d, (3,k)2--Id fork-l,2,3;

77+3,7-0 for#u.

Clearly R() is self-adjoint for all E3. Itseigenvalues are A+(II) and A_(II), where

A+ (r) -t- V/r2 +

(cf. [9]). Each eigenvalue has multiplicity 2 for allE 3. Let S+(), S_() be the spectral projections

of R() corresponding to A+(II) and A_(I) resp.Then we have

U- U+ / U_, U+ S+(D)U, U_ S_(D)U

with the equations

0i--U+ A+( DI)U+,0

i-- U_ A_(IDI)U_

U+(t--O) --S+(D)UI

U_(t--O)--S_(D)UI.

Since A+(r), -A_(r) satisfy the assumption ofCorollary 2.1, we obtain the estimate (3.2a) (forn 3 with the equal sign replaced by "<") for all 4components of U+ and U_. Proceeding as for thewave equation gives for all x0 3.

2iN(x0, t)12dt Igll, (3.8)

which proves the ’regularizing’ effect of the Dirac

equation from H([3) (for the initial data) to

L(R3x;L2(t) ). Also the term involving the

singularity at x x0 in (3.2a) can be computed.We have

S+(c)- I+/-V/I{12 +1and thus"

IX XOI<__ const IUII

I(x-xo). 2

) dxdtX0

for 1,2,3,4,U+/-,3) T.

where U:k (U_+_, 1, U-+-,2, U:t:,2,

Acknowledgement

This research was supported by the GermanDAAD program PROCOPE titled Verallgemei-nerte Halbleitermodelle. The first and the secondauthor were also supported by grant Nr. MA1662/1-2 and 2-2 of the ’Deutsche Forschungsge-meinschaft’ and by grant Nr. 315/PPP/ru-ab ofthe German ’DAAD’. The third author was part-ially supported by TMR project HCL Nr.ERBFMRXCT960033.

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[3] Colin, T. (1994). Smoothing effects for dispersive equa-tions via a generalized Wigner transform, SIAM J. Math.Anal., 25, 1622-1641.

[4] Constantin, P. and Saut, J. C. (1988). Local smoothingproperties of dispersive equations, J. Amer. Math. Soc., 1,413 446.

[5] Constantin, P. and Saut, J. C. (1989). Local smoothingproperties of Schr6dinger equations, Indiana Univ. Math.Journal, 38, 791 810.

[6] Cazenave, T. (1993). An introduction to nonlinearSchr6dinger Equations, Second Edition, Textos deM6todos Matemhticas 26, Universidad Federal do Rio deJaneiro.

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[8] Gasser, I. and Markowich, P. A. (1997). QuantumHydrodynamics, Wigner Transforms and the ClassicalLimit, Asymptotic Analysis, 14, 97-116.

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Authors’ Biographies

I. Gasser received his Ph.D. in Mathematics fromthe Technical University of Berlin (Germany) in1996. He is currently an Assistant Professor at theInstitut ffir Angewandte Mathematik at the Uni-versity of Hamburg (Germany). His fields ofinterest are partial differential equations, semicon-ductor models and kinetic theory.

P. A. Markowich received his Ph.D. in AppliedMathematics from the Technical University ofVienna (Austria) in 1980. He was Professor ofMathematics at the Technical University of Berlin(Germany) from 1989 to 1998 and is currentlyProfessor of Mathematics at the University of Linz(Austria). His fields of interest are partial differ-ential equations and kinetic theory.

B. Perthame completed his these d’etat at theUniversity Paris Dauphine in 1987. He became Pro-fessor ofMathematics at the University d’Orleans in1988. He is currently Professor of Mathematics atthe Ecole Normale Superieure in Paris. His mainfields of interest are partial differential equations,their applications and numerical analysis.

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Dispersion Lemmas Revisiteddownloads.hindawi.com/archive/1999/081341.pdfC6"2fl f [VuI2 o)(l+l/oe) dxdt. with ca > 0. Since (O/Oxl) (XJ(1 + L([n), the treatment ofthe term which